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Appendix II.


Graphical Calculus.—Let a curve be drawn with given
ordinates and abscissae, and be called a primitive curve. A
second curve may be constructed to the same base, which will
shew the gradually increasing value of the area under the first
curve, as we travel say from left to right; and this second curve
we shall call the sum curve. It represents the integration or
summation of the first curve. A third curve can next be drawn
shewing the rate of rise or fall of the primitive curve, and this we
shall term the rate- or slope-curve, being simply a graphic differ-
entiation of the first curve.

Let the primitive curve A B, Fig. 814, rise from j^ tojy2 while

the abscissae change from.% to #2.    Then y increases jy2 —y\ units
of y for ^2 - #! units of ^?, or

Rate of growth       | _ y% - yl = d _

of j, for one unit of x }


This is the mean rate of growth between a and e, and may be
assumed to occur aty; the midway point. The instantaneous rate
may be found by supposing b to gradually become indefinitely
small, when the line a e will finally assume a position tangential to
curve A B, at the point considered. Hence the rate of growth of
a curve is always shewn by the trigonometrical tangent, or the
slope, of a line drawn tangential to the curve.

To draw the Sum Curve, that is, to find area A B c D under